Subgroups — #2081

Order 128.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#430	C₂	1	1	yes	maximal, characteristic, complemented
2	#418	C₂	1	1	yes	maximal, characteristic
3	#418	C₂	1	1	yes	maximal, characteristic
4	C₂×#100	C₂	1	1	yes	maximal, characteristic
5	#344	C₂	1	1	yes	maximal, characteristic, complemented
6	#344	C₂	1	1	yes	maximal, characteristic, complemented
7	H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄	C₂²	1	1	yes	characteristic
9	C₂×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×M₁₆	C₂²	1	1	yes	characteristic
13	#117	C₂²	1	1	yes	characteristic
14	#117	C₂²	1	1	yes	characteristic
15	#108	—	2	2	no	—
16	#102	—	2	2	no	—
17	#102	—	2	2	no	—
18	#100	—	2	2	no	—
19	C₂⁴	—	2	1	no	—
20	C₂×D₄	—	2	1	no	—
21	C₂×D₄	—	2	1	no	—
22	C₂×D₄	—	2	1	no	—
23	C₂×D₄	—	2	1	no	—
24	C₂×D₄	—	2	1	no	—
25	C₂×D₄	D₄	1	2	yes	—
26	C₂²×C₄	C₂³	1	1	yes	characteristic
27	M₁₆	—	2	1	no	—
28	M₁₆	—	2	1	no	—
29	M₁₆	—	2	1	no	—
30	M₁₆	—	2	1	no	—
31	M₁₆	—	2	1	no	—
32	M₁₆	—	2	1	no	—
33	M₁₆	—	2	1	no	—
34	M₁₆	—	2	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	D₄	1	2	yes	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₂²⋊C₄	—	2	1	no	—
42	C₂²⋊C₄	—	2	1	no	—
43	C₄²	—	2	1	no	—
44	C₄²	—	2	1	no	—
45	C₂³	—	2	2	no	—
46	C₂³	—	4	1	no	—
47	C₂³	—	4	1	no	—
48	C₂³	—	8	2	no	—
49	C₂³	#41	1	1	yes	characteristic
50	D₄	—	2	2	no	—
51	D₄	—	2	2	no	—
52	D₄	—	4	1	no	—
53	D₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	4	1	no	—
56	C₂×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄	#41	1	1	yes	characteristic
59	C₂×C₄	#41	1	1	yes	characteristic
60	C₂×C₄	#41	1	1	yes	characteristic
61	C₂×C₄	#41	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	4	1	no	—
64	C₈	—	4	1	no	—
65	C₈	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	2	no	—
68	C₂²	—	8	2	no	—
69	C₂²	—	8	2	no	—
70	C₂²	—	8	2	no	—
71	C₂²	#123	1	1	yes	characteristic
72	C₂²	#124	1	1	yes	characteristic
73	C₂²	#126	1	1	yes	characteristic
74	C₄	—	8	1	no	—
75	C₄	—	8	1	no	—
76	C₄	#121	1	1	yes	characteristic
77	C₄	#123	1	1	yes	characteristic
78	C₄	#123	1	1	yes	characteristic
79	C₄	#126	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	2	1	no	—
82	C₂	—	2	1	no	—
83	C₂	—	8	2	no	—
84	C₂	#397	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#397
commutator, Frattini, cosocle	C₂²×C₄	C₂³
prime radical	C₂²×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #430	C₂	semidirect
5. #344	C₂	semidirect
6. #344	C₂	semidirect
7. H(3,ℤ/4)	C₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#430	2
#418	2
C₂×#100	2
#344	2
H(3,ℤ/4)	2